Let us consider first the distinctions and relations between mathematics, arithmetic, and numeracy.
Mathematics and arithmetic
The related sciences that collectively fall under the term mathematics are concerned with the study of number, quantity, shape and space, and their interrelationships. Mathematics includes algebra, calculus, arithmetic geometry, analytical geometry, chaos theory, number theory and set theory. Arithmetic is based on numerical calculations and quantitative number theory. Arithmetic requires an understanding of basic numerical law and the rules that govern the four basic operations of addition, subtraction, multiplication, and division. Arithmetic also does decoding and manipulating symbols. The basic skills of arithmetic are an essential foundation to the learning of more complex mathematics.
Numeracy or quantitative literacy is the ability to reason with numbers and other mathematical concepts. A numerically literate person can apply different aspects of mathematics to understand, predict, and control routine events in daily life. Aspects of numeracy include number sense, operation sense, computation, measurement, geometry, probability and statistics. The Combination of words “numeracy literacy” was coined in 1959 by the UK committee on Education, presided over by British economist Sir Geoffrey Crowther. Innumeracy is a lack of numeracy.
In simple terms the number sense is an intuitive understanding of numbers. A person with number sense would understand the magnitude, relationships of numbers. He/she would know how the numbers are affected by operations like addition, subtraction and multiplication. Development of number sense helps a child learn to solve problems conceptually rather than procedurally. Specific skills and topics related to number sense include place value, mental arithmetic, and estimation.
Number sense is a biologically based nonverbal system. It is genetically determined and biologically conserved across species. Many animal species as well as human infants are able to discriminate smaller from larger quantity, order a series of relative quantities (e.g. 3, 1, 4, 2 à 1, 2, 3, 4), and manifest a rudimentary nonverbal system for counting. There is a language-based secondary system capable of representing numbers exactly. This system is developed later just like the development of language.
Place value: In our decimal number system, the value of a digit depends on its place, or position, in the number. Each place has a value of 10 times the place to its right. A number in standard form is separated into groups of three digits using commas. Each of these groups is called a period.
Mental arithmetic or mental calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper. People use mental calculation when computing tools are not available, when it is faster than other means of calculation (for example, conventional methods as taught in educational institutions), or in a competition context. Mental calculation often involves the use of specific techniques devised for specific types of problems.
Estimation: When checking the mental calculation, it is useful to think of it in terms of scaling. For example, when dealing with large numbers, say 1531 × 19625, estimation instructs you to be aware of the number of digits expected for the final value. A useful way of checking is to estimate. 1531 is around 1500, and 19625 is around 20000, so a result of around 20000 × 1500 (30000000) would be a good estimate for the actual answer (30045875). So if the answer has too many digits, you know you've made a mistake.
Mathematics disorder or dyscalculia
Mathematics disorder refers to impairment in the development of arithmetic skills, including computational procedures used to solve arithmetic problems. It involves retrieval of basic arithmetic facts from long-term memory. Mathematics difficulties, mathematics disorder, specific disorder of arithmetic skill, math anxiety, and developmental dyscalculia are different terms used to denote mathematics disorder. These terms are not synonyms; but they are similar terms with subtle differences in their meanings. The difficulty is classed as disorder only when the individual’s performance in arithmetic is substantially below that expected for age, measured intellectual abilities, and education. Also, the impairment must be sufficiently serious to interfere with academic achievement or daily living.
Math learning difficulties occur in children with low IQ. But dyscalculia occurs in people across the whole IQ range. Sufferers often, but not always, also have difficulties with time, measurement, and spatial reasoning.
When compared to the reading disability mathematics disorder is less prevalent. Studies in developed countries have established prevalence ranging from 3 to 11 percent of the school children.
Causes: Psychological factors causing mathematics disorder
The causes of mathematics disorder are unknown, but multiple causative factors have been proposed including psychological, neurological, genetic, and social factors. The social factors include poor teaching and math anxiety. Cognitive psychologist Mark H. Ashcraft, Ph.D. defines math anxiety as a feeling of tension, apprehension, or fear that interferes with math performance.
According to Ashcraft, because math anxiety can cause math avoidance, a dilemma arises. For instance, when a highly math-anxious student performs disappointingly on a math question, it could be due to math anxiety, or the lack of competency in math because of math avoidance. In order to distinguish between real math anxiety and lack of competency specially devised psychological tests have to be applied. (Ashcraft, M. H., & Kirk, E. “The relationships among working memory, math anxiety, and performance.” Journalof Experimental Psychology: General 2001, Vol. 130, No. 2, 224-237)
Defects in development
Number sense discussed earlier is primary factor. Basic quantitative capacities develop further through formal education and require additional cognitive capacities, including short-term working memory and symbolization of number in the language. The child also develops a mental number line.
The mental number line
|Mental number line|
Close your eyes and imagine the numbers 1 through 9 on a line. What does the image that appears in your mind look like? Most people will say that they imagine a horizontal line, with 1 on the left, and an orderly progression to 9 on the right. Naturally, this finding could be an effect of cultural convention in societies that use Arabic numerals, but research indicates that there may be more to it than that. As children in Western cultures grow, they learn to place numbers on a mental number line, with smaller numbers to the left and spaced further apart than the larger numbers on the right.
E.g. 1 2 3 4 5 6 7 8 9 10 11 12 13.
Then the number line changes to become more linear, with small and large numbers the same distance apart.
E.g. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16.
The mental number line is a product of schooling and experience. If the number sense is not inherited the child may lean numbers and acquire a mental number line but the numbers will remain void of the meaning of numerical magnitude. They will have gross mathematics disorder.
Neurological causes of dyscalculia
|Brain area playing major role in number processing|
Neuro-imaging studies have established that lower part of the parietal lobe of the brain plays a dominant role in numerical processing. Developmental lag of or injury to this part of the brain may cause gross dyscalculia.
How to identify dyscalculia?
The commonest feature of mathematics disorder is deficit in calculation fluency. In kindergarten and early elementary school the children shows deficit in counting skill. They fail to understand concepts such as more, less, and equivalence. They fail to recognize numerical values of numerals. They are unable to determine which of a pair of numerals is greater or smaller. They cannot copy numbers or write numbers to dictation. They usually count object more than once or out of order.
In older children i.e. third grade and above, major impairments are evident in rapid retrieval of number facts (e. g. 6 x 7 = 42), and in completing procedures necessary to solve more complex arithmetic problems in addition, subtraction, multiplication and division.
Psycho education about the disorder and its longer-term implecation is an essential first step. Treatment of mathematics disorder is typically conducted within educational settings, by professionals trained in special education or educational psychology.